← all cardsTopic 10 · Oscillations · Equations & the two systems
Speed at any displacement
● VERBATIM — examined word-for-wordThe timing-free workhorse — no t needed, only position. Sign gives direction.
v = ±ω√(x₀² − x²)What this actually means
Given a position, this hands you the speed with no timing information at all. It is energy conservation in disguise: kinetic energy is whatever total energy is left after potential.
The ± is direction: the particle passes each x twice per cycle, once moving each way.
Two-position problems: write the formula at both positions and divide; ω or x₀ cancels and the unknowns fall out without any clock.
Prove it — watch it be true
- Open the v–x ellipse: it is v = ±ω√(x₀² − x²) plotted whole, with no time axis anywhere
- Read the intercepts: ±x₀ on the x-axis where v = 0, and ±ωx₀ on the v-axis where the speed is greatest
- Pick any x between them: the ellipse gives two v values, equal in size and opposite in sign — that is the ± in the formula